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It’s interesting to note that Cantor’s work, which is now conventional undergraduate mathematics, was so controversial that it earned him the enmity of figures like Poincare, Kronecker, and even Wittgenstein– and that religious figures once objected to his proofs related to the multiple sizes of infinity.


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    A few more biographical details are in the Cantor biography of the MacTutor History of Mathematics archive of St Andrews, Scotland. Their bios generally have plenty of good references and are my go-to place for quickly looking up the biography of a mathematician.

    Wittgenstein’s attacks on set theory as “utter nonsense” and worse are nicely summarized in the Stanford Encyclopedia of Philosophy.

    I’m a bit disappointed to see the well-known “disease from which one has discovered” quote mentioned as “usually attributed to Poincaré” without further comment. There’s little doubt that Poincaré never said that. See Jeremy Gray, Did Poincaré say “set theory is a disease”?, Math. Intelligencer 13 (1991), no. 1, 19–22 (paywalled) for details.

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      Minor thing - his name was Georg, not George (and pronounced, I believe, like gay-org).

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        I actually knew that– it was a typo. Glad the mods corrected it.

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        It was an exciting time for mathematics.

        Edit of course he was a Baconian…

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          As far as bizarre, offensive [1] theories about prominent people, the Shakespeare/Bacon hypothesis is benign in comparison to… certain modern examples that were once espoused by public figures.

          ETA: It’s also worth noting that his Baconian obsession occurred after a mental breakdown that may have been influenced by the career-ending hostility with which his work was received.

          [1] The reason I argue that the Bacon theory is offensive, or at least politically incorrect, is that the most common argument is that a man from the supposed lower classes could not be one of the English language’s best writers. There are many counterpoints to this, of course. (1) While William Shakespeare was not from the nobility, he was from a solidly middle-class background, and even rich by the standards of his time, and would have had access to literature. He was nouveau riche, not a starving peasant. (2) Shakespeare was not revered as high culture, much less the greatest writer of the English language, until hundreds of years after his death; for a long time, his plays were considered well-crafted but very crude (as, indeed, some of them are). The suggestion that you’d have to be someone highborn like Bacon to write that well is absurd, but not as malicious as, say, certain modern theories.

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            Yes, the interesting thing about the Baconian theory is not the thing itself, but that it still has legs to this day.

            I seem to remember reading a good overview of it some time ago, but searching the usual place for such things (New York Review) turns up zip. And I don’t have a link to it in my link archive.

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          One error that leapt out at me - it says:

          “The symbol for showing that one set is a member of another set is ⊂”

          That’s wrong. That’s the subset symbol, and I suspect it’s the symbol that’s right, and the English that’s wrong.

          The example is right: P ⊂ N ⊂ Z ⊂ Q

          But these are not members of each other, they are subsets.

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            There is still a strong and active “constructive mathematics” community and finitist logic as well. For example: https://en.wikipedia.org/wiki/Internal_set_theory

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              I hadn’t heard of IST until today but this looks cool. Thanks for sharing it. I definitely enjoy alternative logics (e.g. intuitionist logic) so this looks like it’s worth a read.